   Chapter 8.3, Problem 8ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# In each of 7-14, the relation R is an equivalence relation on set A. Find the distinct equivalence classes of R. X = { a , b , c } and A = P ( X ) . R is defined on A as follows: For all sets v and v P ( X ) , v R v   ⇔     N ( v ) = N ( v ) (This is, the number of elements in u equals the number of elements in v.)

To determine

To find the distinct equivalence classes of R.

Explanation

Given information:

The relation R is an equivalence relation on the set A. Find the distinct equivalence classes of R.

X = { a, b, c } and A = P(X). R is defined on A as follows: For all sets U and V in P(X),

U R X = {a, b, c} and A = P(X). R is defined on A as follows:For all sets U and V in P(X),U R V        N(U)=N(V).

(That is, the number of elements in U equals the number of elements in V .)

Calculation:

X={a,b,c}A=P(X)R={(B,C)A×A|B has the same number of elements as C}

Let us first determine all elements in A, which are thus all subsets of X :

A={ϕ,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}}

Two elements are related if they contain the same number of elements:

R={(ϕ,ϕ),({a},{a}),({a},{b}),({a},{c}),({b},{a}),({b},{b}),({b},{c}),      ({c},{a}),({c},{b}),({c},{c}),({a,b},{a,b}),({a,b},{a,c}),({a,b},{b,c}),      ({a,c},{a,b}),({a,c},{a,c}),({a,c},{b,c}),({b,c},{a,b}),({b,c},{a,c}),       ({ b,c},{ b,c}),({ a,b,c},{ a,b,c})}

Let us next group the ordered pairs in R that have at least one common element (with at least one of the other elements in the group)

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